A089215 Thue-Morse sequence on the integers.

1, 2, 3, 2, 4, 3, 2, 3, 5, 4, 3, 4, 2, 3, 4, 3, 6, 5, 4, 5, 3, 4, 5, 4, 2, 3, 4, 3, 5, 4, 3, 4, 7, 6, 5, 6, 4, 5, 6, 5, 3, 4, 5, 4, 6, 5, 4, 5, 2, 3, 4, 3, 5, 4, 3, 4, 6, 5, 4, 5, 3, 4, 5, 4, 8, 7, 6, 7, 5, 6, 7, 6, 4, 5, 6, 5, 7, 6, 5, 6, 3, 4, 5, 4, 6, 5, 4, 5, 7, 6, 5, 6, 4, 5, 6, 5, 2, 3, 4, 3, 5, 4, 3, 4, 6

Offset: 1

Author

Benoit Cloitre, Dec 10 2003

Keywords

• nonn
• easy

Comments

Sequence is S(infinity) where S(1)={1,2} and S(n+1)=S(n)S'(n) where S'(n) is obtained from S(n) by substituting an element x of S(n) with M(n)-x where M(n)=2+Max{S(n)}.

For comparison, the Thue-Morse sequence on alphabet {1,2} is constructed as follows: S(1)={1,2} and S(n+1)=S(n)S'(n) where S'(n) is obtained from S(n) by substituting an element x of S(n) with 3-x.

Examples

			S(1)={1,2} then M(1)=4 and S'(1)={4-1,4-2}={3,2}. So S(2)={1,2,3,2}. M(2)=5 so S(3)={1,2,3,2}{5-1,5-2,5-3,5-2} and sequence begins 1,2,3,2,4,3,2,3,...
		

Crossrefs

Cf. A001285, A209281.

Programs

PARI

a(n) = n--; my(s=1,h); while((h = n>>s), n=bitxor(n,h); s<<=1); hammingweight(n) + 1; \\ Kevin Ryde, Jun 25 2022

Formula

Sum_{k=1..n} a(k) is asymptotic to C*n*log(n) with C=0.8....

a(n) = A209281(n) + 1. - Kevin Ryde, Jun 25 2022